A generalization of the helly property applied to the cliques of a graph

Abstract

Let p ≥ 1 and q ≥ 0 be integers. A family S of sets is (p,q)-intersecting when every subfamily S' ⊆ S formed by p or less members has total intersection of cardinality at least q. A family F of sets is (p,q)-Helly when every (p,q)-intersecting subfamily F' ⊆ F has total intersection of cardinality at least q. A graph G is a (p, q)- clique-Helly graph when its family of cliques (maximal complete sets) is (P, q)-Helly. According to this terminology, the usual Helly property and the clique-Helly graphs correspond to the case p = 2, q = 1. In this work we present characterizations for (p,q)-Helly families of sets and (p,q)-clique-Helly graphs. For fixed p,q those characterizations lead to polynomial-time recognition algorithms. When p or q is not fixed, it is shown that the recognition of (p,q)-clique-Helly graphs is NP-hard. We also extend further the notions presented, by defining the (p,q, r)-Helly property (which holds when every (p, q)-intersecting subfamily F' ⊆ F has total intersection of cardinality at least r) and giving a way of recognizing (p, q, r)-Helly families in terms of the (p, q)-Helly property.